- Email: [email protected]
Semi-global finite-time output feedback stabilization for a class of large-scale uncertain nonlinear systems
Accepted Manuscript Semi-global finite-time output feedback stabilization for a class of large-scale uncertain nonlinear systems Jun-yong Zhai, Wen-ting Zha, Shu-min Fei PII: DOI: Reference:
S1007-5704(13)00141-X http://dx.doi.org/10.1016/j.cnsns.2013.03.016 CNSNS 2757
To appear in:
Communications in Nonlinear Science and Numer‐ ical Simulation
Received Date: Revised Date: Accepted Date:
5 July 2011 24 March 2013 27 March 2013
Please cite this article as: Zhai, J-y., Zha, W-t., Fei, S-m., Semi-global finite-time output feedback stabilization for a class of large-scale uncertain nonlinear systems, Communications in Nonlinear Science and Numerical Simulation (2013), doi: http://dx.doi.org/10.1016/j.cnsns.2013.03.016
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Semi-global finite-time output feedback stabilization for a class of large-scale uncertain nonlinear systems Jun-yong Zhai ∗, Wen-ting Zha, Shu-min Fei Key Laboratory of Measurement and Control of CSE, Ministry of Education School of Automation, Southeast University, Nanjing, Jiangsu 210096, China
Abstract This paper addresses the problem of semi-global finite-time decentralized output feedback control for large-scale systems with both higher-order and lower-order terms. A new design scheme is developed by coupling the finite-time output feedback stabilization method with the homogeneous domination approach. Specifically, we first design a homogeneous observer and an output feedback control law for each nominal subsystem without the nonlinearities. Then, based on the homogeneous domination approach, we relax the linear growth condition to a polynomial one and construct decentralized controllers to render the nonlinear system semi-globally finitetime stable. Keywords: semi-global; finite-time stable; output feedback; nonlinear systems.
1
Introduction
In this paper, we consider the problem of semi-global finite-time stabilization via output feedback for a class of large-scale uncertain nonlinear systems p
ij + ϕij (x, d(t)), j = 1, · · · , n − 1 x˙ ij = xi,j+1
x˙ in = ui + ϕin (x, d(t)) yi = xi1
(1)
where xi = (xi1 , · · · , xin )T ∈ Rn , i = 1, · · · , m, x = (x1 , · · · , xm )T ∈ Rm×n , ui ∈ R, yi ∈ R are system states, control input and output, respectively. The powers pij , j = 1, · · · , n − 1 and pin = 1 are odd integers. ϕij (·)’s are uncertain nonlinear functions of all the states with bounded disturbance d(t). The problem of output feedback stabilization of (1) has attracted a great deal of attention from the nonlinear control community. The existing output feedback stabilizer does not hold for all types of nonlinearities owing to the lack of nonlinear version of separation principle [1]. Therefore certain conditions are required for most of the existing global results. Among the different kinds of assumptions, one common condition is that the unmeasurable states cannot be associated with the uncertainties. To deal with the case when the unmeasurable states are associated with the uncertainties, a feedback domination design method was proposed in [2]. Later, the works ∗
Corresponding author. Email: [email protected]
1
[3, 4, 5] have solved the problem of global output feedback stabilization under a higher-order growth condition by employing the homogeneous domination approach. In [6] we considered the problem of global finite-time stabilization for upper-triangular systems with unknown output gain. To further relax the aforementioned conditions imposed on global output feedback stabilization, in this paper we pursue a less ambitious control objective, i.e. semi-global output feedback stabilization. It has been shown that the restrictive conditions assumed in the global output feedback stabilization problem can be relaxed for semi-global output stabilization. For example, the works [7], [8], [9] and [10] achieved the semi-global output feedback stabilization of feedback linearizable (at least partially) systems. In [11], it was shown that semi-global output feedback stabilization was achievable for uniformly completely observable and state feedback stabilizable systems. The work [12] explicitly constructed a linear output feedback controller to semi-globally exponentially stabilize a class of nonlinear systems under less restrictive conditions. One novelty of the design method is that the observer and controller are linear and independent of the higher-order nonlinearities and the mismatched uncertainties. The paper [13] considered a class of systems in which the subsystem for each output has a triangular dependence on the states of that subsystem itself, and the overall system has a block triangular form for each subsystem. In [14], semi-global robust stabilization was achieved for a class of MIMO nonlinear systems which do not necessarily have a well-defined relative degree nor need to be affine in the control inputs, but exhibit certain triangular structure. In [15], we considered the problem of semi-global stabilization by a series of linear output feedback stabilizers. The work [16] presented the semi-global output feedback stabilization for a class of nonlinear systems with unknown output gains. The problem of output tracking control was addressed for a class of nonlinear strict-feedback form systems in the presence of nonlinear uncertainties, external disturbances, unmodelled dynamics and unknown control coefficients in [17]. Most of the existing works only consider the asymptotic output feedback stabilizer that makes the trajectories of the system converge asymptotically to the equilibrium as the time goes to infinity. Our recent work [18] considered the problem of semi-global finite-time stabilisation by output feedback for a class of uncertain nonlinear systems with both higher-order and lower-order terms. In this paper, we focus on the problem of using output feedback to semi-globally stabilize a class of large-scale nonlinear systems in a finite time, namely, we are interested in designing the decentralized output feedback controllers that will render the trajectories of the closed-loop systems convergent to the origin in a finite time. The large-scale systems have more states and the unmeasurable states interconnected, which make the problem more complicated. A new output feedback design scheme will be developed by coupling the finite-time output feedback stabilization method with the homogeneous domination approach. Specifically, we will first design a homogeneous observer and an output feedback control law for each nominal subsystem. Then, by homogeneous domination approach, a scaling gain will be introduced to render the whole interconnected systems semi-globally finite-time stable. The rest of the paper is organized as follows: In Section 2, we first solve the problem of output feedback stabilization for each nominal subsystem by designing a homogeneous observer and controller. Then, we introduce a scaling gain into the proposed observer and controller, and appropriately adjust the gain to dominate the nonlinearities. An example is included in Section 3 to illustrate the effectiveness of the proposed decentralized output feedback controllers. Our conclusion is in Section 4.
2
2
Main results
In this section, we will employ the homogeneous domination approach to construct decentralized output feedback controllers for system (1). Specifically, we will first construct the homogeneous output feedback controller for each nominal subsystem without considering the nonlinearities ϕij . Then, we will utilize a scaling gain in the controller and observer to dominate the nonlinearities. Firstly, we construct an output feedback stabilizer for the following nominal system pi z˙i = zi+1 , i = 1, · · · , n − 1, z˙n = v, y = z1
(2)
where z = (z1 , · · · , zn )T ∈ Rn , v ∈ R, y ∈ R are system states, control input and output, respectively. Using the approach in [19, 20], we design a homogenous output feedback stabilizer for (2) which is described in the following lemma. Lemma 2.1 For any constant τ¯ ∈ − 1+ n−11(p ···p ) , 0 , there exist constants ki > 0, i = 1, · · · , n− s=1
1
s
1 and βj > 0, j = 1, · · · , n such that the homogeneous output feedback controller η˙ 2 = −k1 zˆ2p1 ,
zˆ2p1 = (η2 + k1 y)r2 p1 /r1
p
p
η˙ i = −ki−1 zˆi i−1 , zˆi i−1 = (ηi + ki−1 zˆi−1 )ri pi−1 /ri−1 , i = 3, · · · , n r +¯τ 1/r 1/r 1/r 1/r 1/r v = −βn zˆn1/rn + βn−1n (ˆ zn−1n−1 + · · · + β2 3 (ˆ z2 2 + β1 2 y) · · · ) n
(3)
with the constants ri ’s defined as r1 = 1, ri + τ¯ = ri+1 pi , i = 1, · · · , n
(4)
renders system (2) globally finite-time stable. Proof: The proof is very similar to the one [20, Theorem 2.1] with some modifications. For the sake of space, the detailed proof is omitted here. Remark 2.1 It should be pointed out that the output feedback controller (3) is only continuous due to the presence of the powers ri pi−1 /ri−1 , i = 2, · · · , n, which are less than one by (4) since τ¯ < 0. As a result, the closed-loop system (2) and (3) is not locally Lipschitz. So the uniqueness of the solution of system (2) and (3) is not guaranteed. Fortunately, as shown in the work [21], the existence of the solution can still be guaranteed for a continuous system without Lipschitz condition. Denoting X := [z1 , · · · , zn , η2 , · · · , ηn ]T , it is straightforward to verify that the closed-loop system (2) and (3), which can be rewritten as the following compact form T X˙ = F (X) = z2p1 , · · · , znpn−1 , v, fn+1 , · · · , f2n−1 , (5) where fn+1 = η˙ 2 , fn+2 = η˙ 3 , · · · , f2n−1 = η˙ n . Moreover, by choosing the dilation weight (refer to [22] for details) Δ = (r1 , · · · , rn , r1 , · · · , rn−1 ) f or z1 ,··· ,zn
= (1, · · · ,
f or η2 ,··· ,ηn
τ τ 1 + (1 + p1 + · · · + p1 · · · pn−3 )¯ 1 + (1 + p1 + · · · + p1 · · · pn−2 )¯ , 1, · · · , ) p1 p2 · · · pn−1 p1 p2 · · · pn−2 f or z1 ,··· ,zn
f or η2 ,··· ,ηn
3
it is clear that system (5) is homogeneous of degree τ¯ with respect to Δ according to the Definition A.2. Since system (5) is globally finite-time stable, by Theorem 2 in [23], there exists a Lyapunov function V (X) with homogeneous of degree 2, such that ∂V τ V˙ (X)|(5) = F (X) ≤ −d1 X2+¯ Δ ∂X
where d1 > 0 and XΔ =
2n−1 2/ri . i=1 |Xi |
(6)
In addition, by Lemma A.3
∂V /∂Xi ≤ d2 X2−ri , d2 > 0. Δ
(7)
Together with the homogeneous controller and observer established in Lemma 2.1, we are ready to use the homogeneous domination approach to semi-globally stabilize (1) via output feedback under the following assumption. r +τ 1 , 0 satisfying ijrlk ≤ Assumption 2.1 There exists any constant τ ∈ − 1+max n−1 1≤i≤m s=1 (pi1 ···pis ) j−1 j−2 qij +1 ml,k s=1 (pi1 · · · pis ) − s=1 (pl1 · · · pls ) > 0, for all 1 ≤ l, i ≤ m, 2 ≤ k ≤ j ≤ n such i,j < qlk and that rij +τ rij +τ rij +τ rij +τ |ϕij | ≤ aij (yi ) (|x11 | r11 + · · · + |x1j | r1j ) + · · · + (|xm1 | rm1 + · · · + |xmj | rmj )
+ bij (yi ) |ϕi1 | ≤
j m
l=1 k=2 ai1 (yi )|xi1 |1+τ
l,k
|xlk |mi,j
(8)
where ri1 = 1, rij = (ri,j−1 +τ )/pi,j−1 , j = 2, · · · , n and qi1 = 0, qij = (qi,j−1 +1)/pi,j−1 , j = 2, · · · , n. aij (yi ) ≥ 0, bij (yi ) ≥ 0 are smooth functions. Remark 2.2 Assumption 2.1 is more general than the assumptions imposed in [18, 24]. It can be easily concluded from (8) that assumption in [24] is a special case of Assumption 2.1 with pij = 1 and bij (·) = 0. In addition, compared to [18], Assumption 2.1 is also more general. It can be seen that assumption in [18] is a special case of Assumption 2.1 with pij = 1 and m = 1. For simplicity, in this paper we assume τ = −qi /pi with qi an even integer and pi an odd integer. Based on this, rij will be odd in both denominator and numerator. Note that a similar result can be achieved for a real number τ as shown in Remark 2.3. Theorem 2.1 Under Assumption 2.1, the problem of semi-global output feedback stabilization for uncertain nonlinear system (1) can be solved by a series of homogeneous output feedback stabilizers. Proof: The output feedback stabilizers are constructed by introducing a scaling gain into the output feedback controller obtained in Lemma 2.1. Then we shall show that for any given domain of the initial condition, all the trajectories starting from the domain will converge to the origin in a finite time.
4
First, we introduce the change of coordinates xij ui zij = qij , i = 1, · · · , m, j = 1, · · · , n and νi = q +1 L L in
(9)
where L ≥ 1 is a constant to be determined later. By construction rij = τ qij + 1/(pi1 pi2 · · · pi,j−1 ), the system (1) can be rewritten as ϕij (·) , j = 1, · · · , n − 1 Lqij ϕin (·) z˙in = Lνi + q L in yi = zi1 p
ij + z˙ij = Lzi,j+1
(10)
Next, we construct the observer with the scaling gain L pi1 pi1 η˙ i2 = −Lki1 zˆi2 , zˆi2 = (ηi2 + ki1 yi )ri2 pi1 /ri1 p
p
η˙ ij = −Lki,j−1 zˆiji,j−1 , zˆiji,j−1 = (ηij + ki,j−1 zˆi,j−1 )rij pi,j−1 /ri,j−1 , j = 3, · · · , n
(11)
where kij > 0, i = 1, · · · , m, j = 1, · · · , n − 1 are the gains selected by Lemma 2.1. In addition, we design ui using the same construction of (3), specifically, ui = Lqin +1 νi , r +τ 1/r 1/rin 1/ri,n−1 1/r 1/r 1/r νi = −βin zˆin in + βi,n−1 (ˆ zi,n−1 + · · · + βi2 i3 (ˆ zi2 i2 + βi1 i2 yi ) · · · ) in .
(12)
For each subsystem, similar to (5), we can obtain Zi = (zi1 , · · · , zin , ηi2 , · · · , ηin )T , p
pi1 Fi (Zi ) = (zi2 , · · · , zini,n−1 , νi , fi,n+1 , · · · , fi,2n−1 )T
(13)
where fi,n+1 = η˙ i2 , · · · , fi,2n−1 = η˙ in . Now the closed-loop system (10)-(12) can be written as ϕi2 ϕin T Z˙ i = LFi (Zi ) + ϕi1 , q , · · · , q , ¯0 (14) L i2 L in where ¯0 is a vector of zeros with suitable dimension. By Lemma 2.1, we can design the gains kij and βij such that
Z˙ i = Fi (Zi )
globally asymptotically stable. Moreover, according to Lemma A.3, there is a Lyapunov function Vi (Zi ) for each subsystem with homogeneous degree 2, such that ∂Vi Fi (Zi ) ≤ −ci Zi 2+τ Δi ∂Zi
(15)
where ci > 0 is a constant and the dilation weight Δi = (ri1 , · · · , rin , ri1 , · · · , ri,n−1 ). f or zi1 ,··· ,zin
f or ηi2 ,··· ,ηin
It can be seen that for each subsystem ∂Vi (Zi ) ∂Vi (Zi ) ϕi2 ϕin T V˙ i (Zi )|(14) = L Fi (Zi ) + ϕi1 , q , · · · , q , ¯0 ∂Zi ∂Zi L i2 L in ϕin T ∂Vi (Zi ) ϕi2 ≤ −Lci Zi 2+τ ϕi1 , q , · · · , q , ¯0 . Δi + ∂Zi L i2 L in
(16)
5
Next, we decide the gain L to guarantee the attractivity for a given domain. To this end, for
an arbitrarily large number M > 1, define compact sets N = η| η ≤ M, η ∈ R2n−1 and
Ω = (Z1 , Z2 , · · · , Zm )|V (Z1 , Z2 , · · · , Zm ) ≤ S, S = max V (Z1 , Z2 , · · · , Zm ) Zi ∈N
m
where V (Z1 , Z2 , · · · , Zm ) = i=1 Vi (Zi ) is the Lyapunov function for the whole system. Clearly, Ω is a nonempty compact set and in this set (Z1 , Z2 , · · · , Zm ) are bounded. According to Assumption 2.1, we have j m
ϕij (·)
|zlk Lqlk |
≤ aij (yi ) Lqij Lqij
rij +τ rlk
j m |zlk Lqlk |mi,j
l,k
+ bij (yi )
l=1 k=1
l=1 k=2
Lqij
(17)
where bi1 (yi ) = 0. The power of the scaling gain L at the first part in the right-hand side of (17) is qlk rij + τ qlk − qij rlk rij + τ qlk − qij = rlk rlk rlk (1 + qij ) − qlk (rij + τ ) =1− rlk rlk qi,j+1 pij − qlk ri,j+1 pij =1− rlk pij =1− (qi,j+1 rlk − qlk ri,j+1 ). rlk Note that qi1 = 0, qij =
qi,j−1 +1 pi,j−1 ,
(18)
qik can be recursively determined as
qik =
1 1 + ··· + , k ≥ 2. pi1 · · · pi,k−1 pi,k−1
(19)
Owing to ri1 = 1, rij + τ = ri,j+1 pij , one obtains 1+τ , pi1 1+τ τ + , ri3 = pi1 pi2 pi2
ri2 =
and recursively rik =
1+τ τ τ + + ··· + , k ≥ 2. pi1 · · · pi,k−1 pi2 · · · pi,k−1 pi,k−1
(20)
In what follows, two cases will be considered. Case 1: when k ≥ 2, we can obtain qi,j+1 rlk − qlk ri,j+1 1 1 τ τ 1+τ = + ··· + + + ··· + pi1 · · · pij pij pl1 · · · pl,k−1 pl2 · · · pl,k−1 pl,k−1 1 1 1 + τ τ τ − + ··· + + + ··· + pl1 · · · pl,k−1 pl,k−1 pi1 · · · pij pi2 · · · pij pij 1 = (1 + pi1 + pi1 pi2 + · · · + pi1 pi2 · · · pi,j−1 ) (pl1 · · · pl,k−1 )(pi1 · · · pij ) − (1 + pl1 + pl1 pl2 + · · · + pl1 pl2 · · · pl,k−2 ) .
(21) 6
By condition
j−1
s=1 (pi1 · · · pis )
−
j−2
s=1 (pl1 · · · pls )
> 0, one has
qi,j+1 rlk − qlk ri,j+1 > 0, for all 2 ≤ k ≤ j ≤ n, 1 ≤ l, i ≤ m.
(22)
Case 2: when k = 1, one obtains qi,j+1 rlk − qlk ri,j+1 =
1 1 + ··· + > 0. pi1 · · · pij pij
(23)
In summary, it can be concluded from (21) and (23) that qi,j+1 rlk − qlk ri,j+1 > 0, for all 1 ≤ k ≤ j ≤ n, 1 ≤ l, i ≤ m
(24)
which implies that the power of the scaling gain L at the first part in the right-hand side of (17) is less than one. Next, we consider the second part in the right-hand side of (17). The power of the scaling gain rij +τ qij +1 l,k L is ml,k i,j qlk − qij . Owing to rlk ≤ mi,j < qlk and 1 ≤ l, i ≤ m, 2 ≤ k ≤ j ≤ n, one obtains ml,k i,j qlk − qij < 1 which in turn implies that there exists a constant αij < 1 such that for L ≥ 1,
j j m m rij +τ
ϕij (·)
ml,k αij rlk
≤ aij (yi )Lαij i,j |z | + b (y )L |z | ij i lk lk
Lqij
= aij (yi )Lαij
l=1 k=1 j m
|zlk |
rij +τ rlk
+ bij (yi )Lαij
l=1 k=1
Owing to ml,k i,j − dij , such that
rij +τ rlk
l=1 k=2 j m
ml,k i,j −
|zlk |
rij +τ rlk
|zlk |
rij +τ rlk
(25)
l=1 k=2
≥ 0. From (25), we know that in the compact set Ω there is a constant j m
ϕ (·)
rij +τ
ij
αij |zlk | rlk .
qij ≤ dij L L
(26)
l=1 k=1
Recall that for j = 1, · · · , n, ∂Vi /∂Zij is homogeneous of degree 2 − rij , then j m
∂V rij +τ
i
|zlk | rlk
∂Zij
(27)
l=1 k=1
is homogeneous of degree 2 + τ by Lemma A.1. With (26) and (27) in mind, by Lemma A.3, we can find a constant ρij > 0 such that
∂Vi ϕij (·)
≤ ρij Lαij Z2+τ
Δ ∂Zij Lqij
(28)
where Z = (Z1 , Z2 , · · · , Zm )T and Δ = (Δ1 , Δ2 , · · · , Δm ). Substituting (28) into (16) yields ∂Vi ϕi2 ϕin (ϕi1 , q , · · · , q , ¯0)T V˙ i ≤ −Lci Zi 2+τ Δi + i2 ∂Zi L L in n ≤ −Lci Zi 2+τ + ρij Lαij Z2+τ Δi Δ .
(29)
j=1
7
It can be seen that the time derivative for the whole system is V˙ (Z) =
m
V˙ i ≤ −LcZ2+τ Δ +
i=1
n m
ρij Lαij Z2+τ Δ
i=1 j=1 αmax −1
≤ −L(c − ρL
)Z2+τ Δ
(30)
where αmax = max1≤i≤m,1≤j≤n {αij } < 1, c > 0 and ρ > 0 that do not depend on L. Apparently, by choosing a large enough L, the right-hand side of (30) is negative definite. In addition, V is homogeneous of degree 2 with respect to Δ. By Lemma A.3, there is a constant c¯1 > 0, such that V (Z) ≤ c¯1 Z2Δ .
(31)
Since the right-hand side of (30) is homogenous of degree 2 + τ , by Lemma A.3 there is a constant c¯2 > 0 such that V˙ (Z) ≤ −¯ c2 Z2+τ Δ .
(32)
Combining (31) and (32), it can be deduced from (30) that V˙ (Z) + k1 V
2+τ 2
(Z) ≤ 0
(33)
for a positive constant k1 > 0. By Lemma A.2 (k = k1 , α = 2+τ 2 < 1), (33) leads to the conclusion that the closed-loop system (1), (11) and (12) is semi-globally finite-time stable. Remark 2.3 During the proof, it is assumed that the fraction number τ has an even numerator and an odd denominator. This requirement can be easily removed. We are still able to design a homogeneous controller globally stabilizing the system with necessary modification to preserve the sign of function [·]ri /ri−1 . Specifically, for any real number ri /ri−1 > 0, we define [·]ri /ri−1 = sign(·)| · |ri /ri−1 . Moreover, it can be verified that the function [·]ri /ri−1 is C 1 .
3
Numerical example
In this section, we provide an example to show the effectiveness of the proposed method. Example 3.1 ⎧ ⎪ ⎨ x˙ 11 = Σ1 : x˙ 12 = ⎪ ⎩ y1 =
Consider the uncertain nonlinear system 4/5
⎧ ⎪ ⎨ x˙ 21 = x22 1/3 3/5 Σ2 : x˙ 22 = u2 + d(t)x12 + (1 + x221 )x22 ⎪ ⎩ y = x 2 21 (34) is trivial. By choosing τ = −2/5, r11 = 1, r12 = 3/5, r21 = 1, r22 =
x12 + d(t)x11 2/15 1/9 u1 + d(t)x22 ln(1 + x222 ) + x11 x12 x11
where |d(t)| ≤ 1. Clearly, ϕ21 3/5, it can be verified that
4/5
|ϕ11 | = |d(t)x11 | ≤ (1 + |x11 |)|x11 |3/5
(35)
8
satisfies Assumption 2.1. In addition, by mean value theory and Lemma A.4, we have ln(1 + x222 ) ≤
3s2 2s3 + 1 2/3 |x | ≤ |x22 |2/3 ≤ 2|x22 |2/3 22 1 + s3 1 + s3
for a s between [0, |x22 |2/3 ]. It can be verified that |d(t)x22 ln(1 + x222 )| ≤ 2|x22 |5/3 .
(36)
By Lemma A.4, one obtains 2/15 1/9
|x11 x12 | ≤ |x11 |1/5 + |x12 |1/3 . 2,2 = 5/3, which satisfies Define m1,2
r12 +τ r22
< m2,2 1,2 <
q12 +1 q22 .
(37)
Combining (36) and (37), it can be seen
2,2 that ϕ12 satisfies Assumption 2.1. Moreover, by defining m2,2 = 3/5 <
|ϕ22 | ≤ |x12 |
r22 +τ2 r12
q22 +1 q22 ,
we can verify that
m2,2
+ (1 + x221 )x222,2
which implies ϕ22 also satisfies Assumption 2.1. Therefore, according to Theorem 2.1, there are homogeneous output feedback controllers rendering the whole nonlinear system semi-globally finitetime stable. Specifically, the output feedback controllers can be constructed as follows: p11 p11 η˙ 12 = −Lk11 zˆ12 , zˆ12 = (η12 + k11 y1 )3/5 1/r12
u1 = −L2 β12 (ˆ z12
1/r12
+ β11
y1 )1/5
p21 p21 η˙ 22 = −Lk21 zˆ22 , zˆ22 = (η22 + k21 y2 )3/5 1/r22
u2 = −L2 β22 (ˆ z22
1/r22
+ β21
y2 )1/5
(38)
where k11 = 2, β11 = 1, β12 = 4, k21 = 15, β21 = 2, β22 = 6 and L = 8. In the simulation, it is assumed that d(t) = sin(t). The simulation result shown in Fig.1 demonstrates the semiglobal finite-time stability property of the closed-loop system (34) and (38) for the initial condition [x11 (0), x12 (0), η12 (0), x21 (0), x22 (0), η22 (0)] = [−1, 3, 0, 2, 1, 0].
0.5
10 x12
20
x11
1
0 −0.5 −1
−10
0
0.2
0.4 t/s
0.6
−20
0.8
3
0
0.2
0.4 t/s
0.6
0.8
0
0.2
0.4 t/s
0.6
0.8
10 0 x22
2 x21
0
1
−10 −20
0 −1
−30 0
0.2
0.4 t/s
0.6
0.8
−40
Fig. 1 The state trajectories of the closed-loop system (34) and (38). 9
4
Conclusions
This paper discusses the problem of semi-global finite-time output feedback control for large-scale nonlinear systems with both higher-order and lower-order terms. The subsystems interconnected by nonlinearities which are functions of the measurable and unmeasurable states, and their linearization cannot guarantee to be either controllable or observable. A new design scheme is developed by coupling the finite-time output feedback stabilization method with the homogeneous domination approach. The output feedback controllers with the appropriate gain will guarantee that the trajectories starting from the given initial region will be bounded all the time and converge to the origin in a finite time.
Acknowledgments This work was supported in part by National Natural Science Foundation of China (61104068, 61273119), Natural Science Foundation of Jiangsu Province (BK2010200), Research Fund for the Doctoral Program of Higher Education of China (20090092120027, 20110092110021), China Postdoctoral Science Foundation Funded Project (2012M511176) and the Fundamental Research Funds for the Central Universities.
Appendix The appendix collects some basic definitions and useful lemmas which serve as the basis for main results. Definition A.1 [25] Consider the following autonomous system x˙ = f (x),
with f (0) = 0, x ∈ D, x(0) = x0
(A1)
where f : D → Rn is continuous on an open neighborhood D ⊆ Rn of the origin. The zero solution of (A1) is finite-time convergent if there are an open neighborhood U ⊆ D of the origin and a function Tx : U\{0} → (0, ∞), such that ∀x0 ∈ U, the solution trajectory x(t, x0 ) of (A1) starting from the initial point x0 ∈ U\{0} is well-defined and unique in forward time for t ∈ 0, Tx (x0 ) , and limt→T (x0 ) x(t, x0 ) = 0. Then, T (x0 ) is called the settling time. The zero solution of (A1) is finite-time stable if it is Lyapunov stable and finite-time convergent. When U = D = Rn , the zero solution is said to be globally finite-time stable. Definition A.2 [22] For real numbers ri > 0, i = 1, · · · , n and fixed coordinates (x1 , · · · , xn ) ∈ Rn , • the dilation Δε (x) is defined by Δε (x) = (εr1 x1 , · · · , εrn xn ), ∀ε > 0, with ri being called as the weights of the coordinates. For simplicity, we define dilation weight Δ = (r1 , · · · , rn ). • a function V ∈ C(Rn , R) is said to be homogeneous of degree τ if there is a real number τ ∈ R such that ∀x ∈ Rn \ {0}, V (Δε (x)) = ετ V (x1 , · · · , xn ).
10
• a vector field f ∈ C(Rn , Rn ) is said to be homogeneous of degree τ if there is a real number τ ∈ R such that for i = 1, · · · , n ∀x ∈ Rn \ {0}, fi (Δε (x)) = ετ +ri fi (x1 , · · · , xn ). n p/ri 1/p , ∀x ∈ Rn , for a constant • a homogeneous p-norm is defined as xΔ,p = i=1 |xi | p ≥ 1. For simplicity, we choose p = 2 and write xΔ for xΔ,2 . Lemma A.1 [22] Let f be a continuous vector field on Rn such that the origin is a locally asymptotical stable equilibrium point. Assume that f is homogeneous of degree k for some r ∈ (0, +∞)n . Then, for any p ∈ N ∗ and any m > p · max{ri }, there exists a strict Lyapunov function V for system (A1), which is homogeneous of degree m and of class C p . As a direct consequence, the time derivative V˙ is homogeneous of degree m + k. Lemma A.2 [25] For a continuous system x˙ = f (x), f (0) = 0, x ∈ Rn
(A2)
suppose that there exists a C 1 positive definite and proper function V : Rn → R and real numbers k > 0 and α ∈ (0, 1) such that V˙ + kV α is negative semi-definite. Then the origin is a finite-time stable equilibrium of (A2). Lemma A.3 [22] Suppose V : Rn → R is a homogeneous function of degree τ with respect to the dilation weight Δ. Then the following hold: (i) ∂V /∂xi is homogeneous of degree τ − ri with ri being the homogeneous weight of xi ; (ii) there is a constant c¯ such that V (x) ≤ c¯xτΔ . Moreover, if V (x) is positive definite, then there exists a constant c > 0, such that V (x) ≥ cxτΔ . Lemma A.4 Suppose c and d are two positive real numbers. Given any real-valued function γ(x, y) > 0, the following inequality holds |x|c |y|d ≤
c c d γ(x, y)|x|c+d + γ − d (x, y)|y|c+d . c+d c+d
(A3)
References [1] Mazenc F, Praly L, Dayawansa WP. Global stabilization by output feedback: examples and counterexamples. Systems and Control Letters 1994;23(2):119–125. [2] Qian CJ, Lin W. Output feedback control of a class of nonlinear systems: a nonseparation principle paradigm. IEEE Trans. Automat. Control 2002;47(10):1710–1715. [3] Qian CJ. A homogeneous domination approach for global output feedback stabilization of a class of nonlinear systems. In Proceedings of 2005 American Control Conference, Portland, OR, USA, pages 4708–4715, June 2005. [4] Andrieu V, Praly L, Astolfi A. Homogeneous approximation and recursive observer design and output feedback. SIAM J Control and Optimization 2008;47(4):1814–1850. 11
[5] Andrieu V, Praly L, Astolfi A. High gain observers with updated gain and homogeneous correction terms. Automatica 2009;45(2):422–428. [6] Zhai JY, Qian CJ. Global finite-time stabilization via output feedback for upper-triangular systems with unknown output gain. In Proceeding of 49th IEEE Conference on Decision and Control, Atlanta, USA, pages 4096–4101, December 2010. [7] Esfandiari F, Khalil H. Output feedback stabilization of fully linearizable systems. International Journal of Control 1992;56(5):1007–1037. [8] Isidori A. Nonlinear control systems (3rd edition). Springer-Verlag: Berlin;1995. [9] Khalil H, Esfandiari F. Semi-global stabilization of a class of nonlinear systems using output feedback. IEEE Trans. Automat. Control 1993;38(9):1412–1415. [10] Lin Z, Saberi A. Robust semi-global stabilization of minimum-phase input-output linearizable systems via partial state and output feedback. IEEE Trans. Automat. Control 1995;40(6):1029– 1041. [11] Teel A, Praly L. Global stabilization and observability imply semi-global stabilization by output feedback. Systems and Control Letters 1994;22(5):313–325. [12] Qian CJ. Semi-global stabilization of a class of uncertain nonlinear systems by linear output feedback. IEEE Trans. Circuits Syst II Express Briefs 2005;52(4):218–222. [13] Shim H, Son YI, Seo JH. Semi-global observer for multi-output nonlinear systems. Systems and Control Letters 2001;42(3):233–244. [14] Lin W, Qian CJ. Semi-global robust stabilization of MIMO nonlinear systems by partial state and dynamic output feedback. Automatica 2001;37(7):1093–1101. [15] Zhai JY, Qian CJ, Ye H. Semi-global output feedback stabilization of a generalized class of MIMO nonlinear systems. ASME Journal of Dynamic Systems, Measurement, and Control 2011;133(6):064501.1–064501.5. [16] Zhai JY, Du HB. Semi-global output feedback stabilization for a class of nonlinear systems using homogeneous domination approach. ISA Transactions 2013;52(2):231–241. [17] Yu Y, Zhong Y. Semi-global robust output tracking for non-linear uncertain systems in strictfeedback form. IET Control Theory and Applications 2012;6(5):751–759. [18] Jia RT, Qian CJ, Zhai JY. Semi-global stabilisation of uncertain non-linear systems by homogeneous output feedback controllers. IET Control Theory and Applications 2012;6(1):165–172. [19] Polendo J, Qian CJ. A generalized homogeneous domination approach for global stabilization of inherently nonlinear systems via output feedback. International Journal of Robust and Nonlinear Control 2007;17:605–629. [20] Zhai JY, Fei SM. Global practical tracking control for a class of uncertain nonlinear systems. IET Control Theory and Applications 2011;5(11):1343–1351. 12
[21] Kurzweil J. On the inversion of Lyapunov’s second theorem on the stability of motion. Ann. Math. Soc. Transl. Ser. 1956;24(2):19–77. [22] Bacciotti A, Rosier L. Liapunov functions and stability in control theory. Springer;2005. [23] Hermes H. Homogeneous coordinates and continuous asymptotically stabilizing feedback controls. Differential equations, Volume 127, Dekker: New York;1991. [24] Polendo J, Qian CJ. Decentralized output feedback control of interconnected systems with high-order nonlinearities. In Proceedings of 2007 American Control Conference, New York City, USA, pages 1479–1484, July 2007. [25] Bhat S, Bernstein D. Finite-time stability of continuous autonomous systems. SIAM J Control and Optimization 2000;38(3):751-766. [26] Bhat S, Bernstein D. Finite time stability of homogeneous systems. In Proceedings of 1997 American Control Conference, Albuquerque, New Mexico, USA, pages 2513–2514, June 1997.
13
Highlights
ŹA new design scheme is developed by coupling the finite-time stabilization theory with the homogeneous domination approach. ŹA controller is constructed for each nominal system by adding one power integrator technique. ŹThe stability and convergence analysis are carried out using the homogeneous systems theory. ŹThe efficiency of the decentralized output feedback stabilizers is demonstrated by a simulation example.
S1007-5704(13)00141-X http://dx.doi.org/10.1016/j.cnsns.2013.03.016 CNSNS 2757
To appear in:
Communications in Nonlinear Science and Numer‐ ical Simulation
Received Date: Revised Date: Accepted Date:
5 July 2011 24 March 2013 27 March 2013
Please cite this article as: Zhai, J-y., Zha, W-t., Fei, S-m., Semi-global finite-time output feedback stabilization for a class of large-scale uncertain nonlinear systems, Communications in Nonlinear Science and Numerical Simulation (2013), doi: http://dx.doi.org/10.1016/j.cnsns.2013.03.016
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Semi-global finite-time output feedback stabilization for a class of large-scale uncertain nonlinear systems Jun-yong Zhai ∗, Wen-ting Zha, Shu-min Fei Key Laboratory of Measurement and Control of CSE, Ministry of Education School of Automation, Southeast University, Nanjing, Jiangsu 210096, China
Abstract This paper addresses the problem of semi-global finite-time decentralized output feedback control for large-scale systems with both higher-order and lower-order terms. A new design scheme is developed by coupling the finite-time output feedback stabilization method with the homogeneous domination approach. Specifically, we first design a homogeneous observer and an output feedback control law for each nominal subsystem without the nonlinearities. Then, based on the homogeneous domination approach, we relax the linear growth condition to a polynomial one and construct decentralized controllers to render the nonlinear system semi-globally finitetime stable. Keywords: semi-global; finite-time stable; output feedback; nonlinear systems.
1
Introduction
In this paper, we consider the problem of semi-global finite-time stabilization via output feedback for a class of large-scale uncertain nonlinear systems p
ij + ϕij (x, d(t)), j = 1, · · · , n − 1 x˙ ij = xi,j+1
x˙ in = ui + ϕin (x, d(t)) yi = xi1
(1)
where xi = (xi1 , · · · , xin )T ∈ Rn , i = 1, · · · , m, x = (x1 , · · · , xm )T ∈ Rm×n , ui ∈ R, yi ∈ R are system states, control input and output, respectively. The powers pij , j = 1, · · · , n − 1 and pin = 1 are odd integers. ϕij (·)’s are uncertain nonlinear functions of all the states with bounded disturbance d(t). The problem of output feedback stabilization of (1) has attracted a great deal of attention from the nonlinear control community. The existing output feedback stabilizer does not hold for all types of nonlinearities owing to the lack of nonlinear version of separation principle [1]. Therefore certain conditions are required for most of the existing global results. Among the different kinds of assumptions, one common condition is that the unmeasurable states cannot be associated with the uncertainties. To deal with the case when the unmeasurable states are associated with the uncertainties, a feedback domination design method was proposed in [2]. Later, the works ∗
Corresponding author. Email: [email protected]
1
[3, 4, 5] have solved the problem of global output feedback stabilization under a higher-order growth condition by employing the homogeneous domination approach. In [6] we considered the problem of global finite-time stabilization for upper-triangular systems with unknown output gain. To further relax the aforementioned conditions imposed on global output feedback stabilization, in this paper we pursue a less ambitious control objective, i.e. semi-global output feedback stabilization. It has been shown that the restrictive conditions assumed in the global output feedback stabilization problem can be relaxed for semi-global output stabilization. For example, the works [7], [8], [9] and [10] achieved the semi-global output feedback stabilization of feedback linearizable (at least partially) systems. In [11], it was shown that semi-global output feedback stabilization was achievable for uniformly completely observable and state feedback stabilizable systems. The work [12] explicitly constructed a linear output feedback controller to semi-globally exponentially stabilize a class of nonlinear systems under less restrictive conditions. One novelty of the design method is that the observer and controller are linear and independent of the higher-order nonlinearities and the mismatched uncertainties. The paper [13] considered a class of systems in which the subsystem for each output has a triangular dependence on the states of that subsystem itself, and the overall system has a block triangular form for each subsystem. In [14], semi-global robust stabilization was achieved for a class of MIMO nonlinear systems which do not necessarily have a well-defined relative degree nor need to be affine in the control inputs, but exhibit certain triangular structure. In [15], we considered the problem of semi-global stabilization by a series of linear output feedback stabilizers. The work [16] presented the semi-global output feedback stabilization for a class of nonlinear systems with unknown output gains. The problem of output tracking control was addressed for a class of nonlinear strict-feedback form systems in the presence of nonlinear uncertainties, external disturbances, unmodelled dynamics and unknown control coefficients in [17]. Most of the existing works only consider the asymptotic output feedback stabilizer that makes the trajectories of the system converge asymptotically to the equilibrium as the time goes to infinity. Our recent work [18] considered the problem of semi-global finite-time stabilisation by output feedback for a class of uncertain nonlinear systems with both higher-order and lower-order terms. In this paper, we focus on the problem of using output feedback to semi-globally stabilize a class of large-scale nonlinear systems in a finite time, namely, we are interested in designing the decentralized output feedback controllers that will render the trajectories of the closed-loop systems convergent to the origin in a finite time. The large-scale systems have more states and the unmeasurable states interconnected, which make the problem more complicated. A new output feedback design scheme will be developed by coupling the finite-time output feedback stabilization method with the homogeneous domination approach. Specifically, we will first design a homogeneous observer and an output feedback control law for each nominal subsystem. Then, by homogeneous domination approach, a scaling gain will be introduced to render the whole interconnected systems semi-globally finite-time stable. The rest of the paper is organized as follows: In Section 2, we first solve the problem of output feedback stabilization for each nominal subsystem by designing a homogeneous observer and controller. Then, we introduce a scaling gain into the proposed observer and controller, and appropriately adjust the gain to dominate the nonlinearities. An example is included in Section 3 to illustrate the effectiveness of the proposed decentralized output feedback controllers. Our conclusion is in Section 4.
2
2
Main results
In this section, we will employ the homogeneous domination approach to construct decentralized output feedback controllers for system (1). Specifically, we will first construct the homogeneous output feedback controller for each nominal subsystem without considering the nonlinearities ϕij . Then, we will utilize a scaling gain in the controller and observer to dominate the nonlinearities. Firstly, we construct an output feedback stabilizer for the following nominal system pi z˙i = zi+1 , i = 1, · · · , n − 1, z˙n = v, y = z1
(2)
where z = (z1 , · · · , zn )T ∈ Rn , v ∈ R, y ∈ R are system states, control input and output, respectively. Using the approach in [19, 20], we design a homogenous output feedback stabilizer for (2) which is described in the following lemma. Lemma 2.1 For any constant τ¯ ∈ − 1+ n−11(p ···p ) , 0 , there exist constants ki > 0, i = 1, · · · , n− s=1
1
s
1 and βj > 0, j = 1, · · · , n such that the homogeneous output feedback controller η˙ 2 = −k1 zˆ2p1 ,
zˆ2p1 = (η2 + k1 y)r2 p1 /r1
p
p
η˙ i = −ki−1 zˆi i−1 , zˆi i−1 = (ηi + ki−1 zˆi−1 )ri pi−1 /ri−1 , i = 3, · · · , n r +¯τ 1/r 1/r 1/r 1/r 1/r v = −βn zˆn1/rn + βn−1n (ˆ zn−1n−1 + · · · + β2 3 (ˆ z2 2 + β1 2 y) · · · ) n
(3)
with the constants ri ’s defined as r1 = 1, ri + τ¯ = ri+1 pi , i = 1, · · · , n
(4)
renders system (2) globally finite-time stable. Proof: The proof is very similar to the one [20, Theorem 2.1] with some modifications. For the sake of space, the detailed proof is omitted here. Remark 2.1 It should be pointed out that the output feedback controller (3) is only continuous due to the presence of the powers ri pi−1 /ri−1 , i = 2, · · · , n, which are less than one by (4) since τ¯ < 0. As a result, the closed-loop system (2) and (3) is not locally Lipschitz. So the uniqueness of the solution of system (2) and (3) is not guaranteed. Fortunately, as shown in the work [21], the existence of the solution can still be guaranteed for a continuous system without Lipschitz condition. Denoting X := [z1 , · · · , zn , η2 , · · · , ηn ]T , it is straightforward to verify that the closed-loop system (2) and (3), which can be rewritten as the following compact form T X˙ = F (X) = z2p1 , · · · , znpn−1 , v, fn+1 , · · · , f2n−1 , (5) where fn+1 = η˙ 2 , fn+2 = η˙ 3 , · · · , f2n−1 = η˙ n . Moreover, by choosing the dilation weight (refer to [22] for details) Δ = (r1 , · · · , rn , r1 , · · · , rn−1 ) f or z1 ,··· ,zn
= (1, · · · ,
f or η2 ,··· ,ηn
τ τ 1 + (1 + p1 + · · · + p1 · · · pn−3 )¯ 1 + (1 + p1 + · · · + p1 · · · pn−2 )¯ , 1, · · · , ) p1 p2 · · · pn−1 p1 p2 · · · pn−2 f or z1 ,··· ,zn
f or η2 ,··· ,ηn
3
it is clear that system (5) is homogeneous of degree τ¯ with respect to Δ according to the Definition A.2. Since system (5) is globally finite-time stable, by Theorem 2 in [23], there exists a Lyapunov function V (X) with homogeneous of degree 2, such that ∂V τ V˙ (X)|(5) = F (X) ≤ −d1 X2+¯ Δ ∂X
where d1 > 0 and XΔ =
2n−1 2/ri . i=1 |Xi |
(6)
In addition, by Lemma A.3
∂V /∂Xi ≤ d2 X2−ri , d2 > 0. Δ
(7)
Together with the homogeneous controller and observer established in Lemma 2.1, we are ready to use the homogeneous domination approach to semi-globally stabilize (1) via output feedback under the following assumption. r +τ 1 , 0 satisfying ijrlk ≤ Assumption 2.1 There exists any constant τ ∈ − 1+max n−1 1≤i≤m s=1 (pi1 ···pis ) j−1 j−2 qij +1 ml,k s=1 (pi1 · · · pis ) − s=1 (pl1 · · · pls ) > 0, for all 1 ≤ l, i ≤ m, 2 ≤ k ≤ j ≤ n such i,j < qlk and that rij +τ rij +τ rij +τ rij +τ |ϕij | ≤ aij (yi ) (|x11 | r11 + · · · + |x1j | r1j ) + · · · + (|xm1 | rm1 + · · · + |xmj | rmj )
+ bij (yi ) |ϕi1 | ≤
j m
l=1 k=2 ai1 (yi )|xi1 |1+τ
l,k
|xlk |mi,j
(8)
where ri1 = 1, rij = (ri,j−1 +τ )/pi,j−1 , j = 2, · · · , n and qi1 = 0, qij = (qi,j−1 +1)/pi,j−1 , j = 2, · · · , n. aij (yi ) ≥ 0, bij (yi ) ≥ 0 are smooth functions. Remark 2.2 Assumption 2.1 is more general than the assumptions imposed in [18, 24]. It can be easily concluded from (8) that assumption in [24] is a special case of Assumption 2.1 with pij = 1 and bij (·) = 0. In addition, compared to [18], Assumption 2.1 is also more general. It can be seen that assumption in [18] is a special case of Assumption 2.1 with pij = 1 and m = 1. For simplicity, in this paper we assume τ = −qi /pi with qi an even integer and pi an odd integer. Based on this, rij will be odd in both denominator and numerator. Note that a similar result can be achieved for a real number τ as shown in Remark 2.3. Theorem 2.1 Under Assumption 2.1, the problem of semi-global output feedback stabilization for uncertain nonlinear system (1) can be solved by a series of homogeneous output feedback stabilizers. Proof: The output feedback stabilizers are constructed by introducing a scaling gain into the output feedback controller obtained in Lemma 2.1. Then we shall show that for any given domain of the initial condition, all the trajectories starting from the domain will converge to the origin in a finite time.
4
First, we introduce the change of coordinates xij ui zij = qij , i = 1, · · · , m, j = 1, · · · , n and νi = q +1 L L in
(9)
where L ≥ 1 is a constant to be determined later. By construction rij = τ qij + 1/(pi1 pi2 · · · pi,j−1 ), the system (1) can be rewritten as ϕij (·) , j = 1, · · · , n − 1 Lqij ϕin (·) z˙in = Lνi + q L in yi = zi1 p
ij + z˙ij = Lzi,j+1
(10)
Next, we construct the observer with the scaling gain L pi1 pi1 η˙ i2 = −Lki1 zˆi2 , zˆi2 = (ηi2 + ki1 yi )ri2 pi1 /ri1 p
p
η˙ ij = −Lki,j−1 zˆiji,j−1 , zˆiji,j−1 = (ηij + ki,j−1 zˆi,j−1 )rij pi,j−1 /ri,j−1 , j = 3, · · · , n
(11)
where kij > 0, i = 1, · · · , m, j = 1, · · · , n − 1 are the gains selected by Lemma 2.1. In addition, we design ui using the same construction of (3), specifically, ui = Lqin +1 νi , r +τ 1/r 1/rin 1/ri,n−1 1/r 1/r 1/r νi = −βin zˆin in + βi,n−1 (ˆ zi,n−1 + · · · + βi2 i3 (ˆ zi2 i2 + βi1 i2 yi ) · · · ) in .
(12)
For each subsystem, similar to (5), we can obtain Zi = (zi1 , · · · , zin , ηi2 , · · · , ηin )T , p
pi1 Fi (Zi ) = (zi2 , · · · , zini,n−1 , νi , fi,n+1 , · · · , fi,2n−1 )T
(13)
where fi,n+1 = η˙ i2 , · · · , fi,2n−1 = η˙ in . Now the closed-loop system (10)-(12) can be written as ϕi2 ϕin T Z˙ i = LFi (Zi ) + ϕi1 , q , · · · , q , ¯0 (14) L i2 L in where ¯0 is a vector of zeros with suitable dimension. By Lemma 2.1, we can design the gains kij and βij such that
Z˙ i = Fi (Zi )
globally asymptotically stable. Moreover, according to Lemma A.3, there is a Lyapunov function Vi (Zi ) for each subsystem with homogeneous degree 2, such that ∂Vi Fi (Zi ) ≤ −ci Zi 2+τ Δi ∂Zi
(15)
where ci > 0 is a constant and the dilation weight Δi = (ri1 , · · · , rin , ri1 , · · · , ri,n−1 ). f or zi1 ,··· ,zin
f or ηi2 ,··· ,ηin
It can be seen that for each subsystem ∂Vi (Zi ) ∂Vi (Zi ) ϕi2 ϕin T V˙ i (Zi )|(14) = L Fi (Zi ) + ϕi1 , q , · · · , q , ¯0 ∂Zi ∂Zi L i2 L in ϕin T ∂Vi (Zi ) ϕi2 ≤ −Lci Zi 2+τ ϕi1 , q , · · · , q , ¯0 . Δi + ∂Zi L i2 L in
(16)
5
Next, we decide the gain L to guarantee the attractivity for a given domain. To this end, for
an arbitrarily large number M > 1, define compact sets N = η| η ≤ M, η ∈ R2n−1 and
Ω = (Z1 , Z2 , · · · , Zm )|V (Z1 , Z2 , · · · , Zm ) ≤ S, S = max V (Z1 , Z2 , · · · , Zm ) Zi ∈N
m
where V (Z1 , Z2 , · · · , Zm ) = i=1 Vi (Zi ) is the Lyapunov function for the whole system. Clearly, Ω is a nonempty compact set and in this set (Z1 , Z2 , · · · , Zm ) are bounded. According to Assumption 2.1, we have j m
ϕij (·)
|zlk Lqlk |
≤ aij (yi ) Lqij Lqij
rij +τ rlk
j m |zlk Lqlk |mi,j
l,k
+ bij (yi )
l=1 k=1
l=1 k=2
Lqij
(17)
where bi1 (yi ) = 0. The power of the scaling gain L at the first part in the right-hand side of (17) is qlk rij + τ qlk − qij rlk rij + τ qlk − qij = rlk rlk rlk (1 + qij ) − qlk (rij + τ ) =1− rlk rlk qi,j+1 pij − qlk ri,j+1 pij =1− rlk pij =1− (qi,j+1 rlk − qlk ri,j+1 ). rlk Note that qi1 = 0, qij =
qi,j−1 +1 pi,j−1 ,
(18)
qik can be recursively determined as
qik =
1 1 + ··· + , k ≥ 2. pi1 · · · pi,k−1 pi,k−1
(19)
Owing to ri1 = 1, rij + τ = ri,j+1 pij , one obtains 1+τ , pi1 1+τ τ + , ri3 = pi1 pi2 pi2
ri2 =
and recursively rik =
1+τ τ τ + + ··· + , k ≥ 2. pi1 · · · pi,k−1 pi2 · · · pi,k−1 pi,k−1
(20)
In what follows, two cases will be considered. Case 1: when k ≥ 2, we can obtain qi,j+1 rlk − qlk ri,j+1 1 1 τ τ 1+τ = + ··· + + + ··· + pi1 · · · pij pij pl1 · · · pl,k−1 pl2 · · · pl,k−1 pl,k−1 1 1 1 + τ τ τ − + ··· + + + ··· + pl1 · · · pl,k−1 pl,k−1 pi1 · · · pij pi2 · · · pij pij 1 = (1 + pi1 + pi1 pi2 + · · · + pi1 pi2 · · · pi,j−1 ) (pl1 · · · pl,k−1 )(pi1 · · · pij ) − (1 + pl1 + pl1 pl2 + · · · + pl1 pl2 · · · pl,k−2 ) .
(21) 6
By condition
j−1
s=1 (pi1 · · · pis )
−
j−2
s=1 (pl1 · · · pls )
> 0, one has
qi,j+1 rlk − qlk ri,j+1 > 0, for all 2 ≤ k ≤ j ≤ n, 1 ≤ l, i ≤ m.
(22)
Case 2: when k = 1, one obtains qi,j+1 rlk − qlk ri,j+1 =
1 1 + ··· + > 0. pi1 · · · pij pij
(23)
In summary, it can be concluded from (21) and (23) that qi,j+1 rlk − qlk ri,j+1 > 0, for all 1 ≤ k ≤ j ≤ n, 1 ≤ l, i ≤ m
(24)
which implies that the power of the scaling gain L at the first part in the right-hand side of (17) is less than one. Next, we consider the second part in the right-hand side of (17). The power of the scaling gain rij +τ qij +1 l,k L is ml,k i,j qlk − qij . Owing to rlk ≤ mi,j < qlk and 1 ≤ l, i ≤ m, 2 ≤ k ≤ j ≤ n, one obtains ml,k i,j qlk − qij < 1 which in turn implies that there exists a constant αij < 1 such that for L ≥ 1,
j j m m rij +τ
ϕij (·)
ml,k αij rlk
≤ aij (yi )Lαij i,j |z | + b (y )L |z | ij i lk lk
Lqij
= aij (yi )Lαij
l=1 k=1 j m
|zlk |
rij +τ rlk
+ bij (yi )Lαij
l=1 k=1
Owing to ml,k i,j − dij , such that
rij +τ rlk
l=1 k=2 j m
ml,k i,j −
|zlk |
rij +τ rlk
|zlk |
rij +τ rlk
(25)
l=1 k=2
≥ 0. From (25), we know that in the compact set Ω there is a constant j m
ϕ (·)
rij +τ
ij
αij |zlk | rlk .
qij ≤ dij L L
(26)
l=1 k=1
Recall that for j = 1, · · · , n, ∂Vi /∂Zij is homogeneous of degree 2 − rij , then j m
∂V rij +τ
i
|zlk | rlk
∂Zij
(27)
l=1 k=1
is homogeneous of degree 2 + τ by Lemma A.1. With (26) and (27) in mind, by Lemma A.3, we can find a constant ρij > 0 such that
∂Vi ϕij (·)
≤ ρij Lαij Z2+τ
Δ ∂Zij Lqij
(28)
where Z = (Z1 , Z2 , · · · , Zm )T and Δ = (Δ1 , Δ2 , · · · , Δm ). Substituting (28) into (16) yields ∂Vi ϕi2 ϕin (ϕi1 , q , · · · , q , ¯0)T V˙ i ≤ −Lci Zi 2+τ Δi + i2 ∂Zi L L in n ≤ −Lci Zi 2+τ + ρij Lαij Z2+τ Δi Δ .
(29)
j=1
7
It can be seen that the time derivative for the whole system is V˙ (Z) =
m
V˙ i ≤ −LcZ2+τ Δ +
i=1
n m
ρij Lαij Z2+τ Δ
i=1 j=1 αmax −1
≤ −L(c − ρL
)Z2+τ Δ
(30)
where αmax = max1≤i≤m,1≤j≤n {αij } < 1, c > 0 and ρ > 0 that do not depend on L. Apparently, by choosing a large enough L, the right-hand side of (30) is negative definite. In addition, V is homogeneous of degree 2 with respect to Δ. By Lemma A.3, there is a constant c¯1 > 0, such that V (Z) ≤ c¯1 Z2Δ .
(31)
Since the right-hand side of (30) is homogenous of degree 2 + τ , by Lemma A.3 there is a constant c¯2 > 0 such that V˙ (Z) ≤ −¯ c2 Z2+τ Δ .
(32)
Combining (31) and (32), it can be deduced from (30) that V˙ (Z) + k1 V
2+τ 2
(Z) ≤ 0
(33)
for a positive constant k1 > 0. By Lemma A.2 (k = k1 , α = 2+τ 2 < 1), (33) leads to the conclusion that the closed-loop system (1), (11) and (12) is semi-globally finite-time stable. Remark 2.3 During the proof, it is assumed that the fraction number τ has an even numerator and an odd denominator. This requirement can be easily removed. We are still able to design a homogeneous controller globally stabilizing the system with necessary modification to preserve the sign of function [·]ri /ri−1 . Specifically, for any real number ri /ri−1 > 0, we define [·]ri /ri−1 = sign(·)| · |ri /ri−1 . Moreover, it can be verified that the function [·]ri /ri−1 is C 1 .
3
Numerical example
In this section, we provide an example to show the effectiveness of the proposed method. Example 3.1 ⎧ ⎪ ⎨ x˙ 11 = Σ1 : x˙ 12 = ⎪ ⎩ y1 =
Consider the uncertain nonlinear system 4/5
⎧ ⎪ ⎨ x˙ 21 = x22 1/3 3/5 Σ2 : x˙ 22 = u2 + d(t)x12 + (1 + x221 )x22 ⎪ ⎩ y = x 2 21 (34) is trivial. By choosing τ = −2/5, r11 = 1, r12 = 3/5, r21 = 1, r22 =
x12 + d(t)x11 2/15 1/9 u1 + d(t)x22 ln(1 + x222 ) + x11 x12 x11
where |d(t)| ≤ 1. Clearly, ϕ21 3/5, it can be verified that
4/5
|ϕ11 | = |d(t)x11 | ≤ (1 + |x11 |)|x11 |3/5
(35)
8
satisfies Assumption 2.1. In addition, by mean value theory and Lemma A.4, we have ln(1 + x222 ) ≤
3s2 2s3 + 1 2/3 |x | ≤ |x22 |2/3 ≤ 2|x22 |2/3 22 1 + s3 1 + s3
for a s between [0, |x22 |2/3 ]. It can be verified that |d(t)x22 ln(1 + x222 )| ≤ 2|x22 |5/3 .
(36)
By Lemma A.4, one obtains 2/15 1/9
|x11 x12 | ≤ |x11 |1/5 + |x12 |1/3 . 2,2 = 5/3, which satisfies Define m1,2
r12 +τ r22
< m2,2 1,2 <
q12 +1 q22 .
(37)
Combining (36) and (37), it can be seen
2,2 that ϕ12 satisfies Assumption 2.1. Moreover, by defining m2,2 = 3/5 <
|ϕ22 | ≤ |x12 |
r22 +τ2 r12
q22 +1 q22 ,
we can verify that
m2,2
+ (1 + x221 )x222,2
which implies ϕ22 also satisfies Assumption 2.1. Therefore, according to Theorem 2.1, there are homogeneous output feedback controllers rendering the whole nonlinear system semi-globally finitetime stable. Specifically, the output feedback controllers can be constructed as follows: p11 p11 η˙ 12 = −Lk11 zˆ12 , zˆ12 = (η12 + k11 y1 )3/5 1/r12
u1 = −L2 β12 (ˆ z12
1/r12
+ β11
y1 )1/5
p21 p21 η˙ 22 = −Lk21 zˆ22 , zˆ22 = (η22 + k21 y2 )3/5 1/r22
u2 = −L2 β22 (ˆ z22
1/r22
+ β21
y2 )1/5
(38)
where k11 = 2, β11 = 1, β12 = 4, k21 = 15, β21 = 2, β22 = 6 and L = 8. In the simulation, it is assumed that d(t) = sin(t). The simulation result shown in Fig.1 demonstrates the semiglobal finite-time stability property of the closed-loop system (34) and (38) for the initial condition [x11 (0), x12 (0), η12 (0), x21 (0), x22 (0), η22 (0)] = [−1, 3, 0, 2, 1, 0].
0.5
10 x12
20
x11
1
0 −0.5 −1
−10
0
0.2
0.4 t/s
0.6
−20
0.8
3
0
0.2
0.4 t/s
0.6
0.8
0
0.2
0.4 t/s
0.6
0.8
10 0 x22
2 x21
0
1
−10 −20
0 −1
−30 0
0.2
0.4 t/s
0.6
0.8
−40
Fig. 1 The state trajectories of the closed-loop system (34) and (38). 9
4
Conclusions
This paper discusses the problem of semi-global finite-time output feedback control for large-scale nonlinear systems with both higher-order and lower-order terms. The subsystems interconnected by nonlinearities which are functions of the measurable and unmeasurable states, and their linearization cannot guarantee to be either controllable or observable. A new design scheme is developed by coupling the finite-time output feedback stabilization method with the homogeneous domination approach. The output feedback controllers with the appropriate gain will guarantee that the trajectories starting from the given initial region will be bounded all the time and converge to the origin in a finite time.
Acknowledgments This work was supported in part by National Natural Science Foundation of China (61104068, 61273119), Natural Science Foundation of Jiangsu Province (BK2010200), Research Fund for the Doctoral Program of Higher Education of China (20090092120027, 20110092110021), China Postdoctoral Science Foundation Funded Project (2012M511176) and the Fundamental Research Funds for the Central Universities.
Appendix The appendix collects some basic definitions and useful lemmas which serve as the basis for main results. Definition A.1 [25] Consider the following autonomous system x˙ = f (x),
with f (0) = 0, x ∈ D, x(0) = x0
(A1)
where f : D → Rn is continuous on an open neighborhood D ⊆ Rn of the origin. The zero solution of (A1) is finite-time convergent if there are an open neighborhood U ⊆ D of the origin and a function Tx : U\{0} → (0, ∞), such that ∀x0 ∈ U, the solution trajectory x(t, x0 ) of (A1) starting from the initial point x0 ∈ U\{0} is well-defined and unique in forward time for t ∈ 0, Tx (x0 ) , and limt→T (x0 ) x(t, x0 ) = 0. Then, T (x0 ) is called the settling time. The zero solution of (A1) is finite-time stable if it is Lyapunov stable and finite-time convergent. When U = D = Rn , the zero solution is said to be globally finite-time stable. Definition A.2 [22] For real numbers ri > 0, i = 1, · · · , n and fixed coordinates (x1 , · · · , xn ) ∈ Rn , • the dilation Δε (x) is defined by Δε (x) = (εr1 x1 , · · · , εrn xn ), ∀ε > 0, with ri being called as the weights of the coordinates. For simplicity, we define dilation weight Δ = (r1 , · · · , rn ). • a function V ∈ C(Rn , R) is said to be homogeneous of degree τ if there is a real number τ ∈ R such that ∀x ∈ Rn \ {0}, V (Δε (x)) = ετ V (x1 , · · · , xn ).
10
• a vector field f ∈ C(Rn , Rn ) is said to be homogeneous of degree τ if there is a real number τ ∈ R such that for i = 1, · · · , n ∀x ∈ Rn \ {0}, fi (Δε (x)) = ετ +ri fi (x1 , · · · , xn ). n p/ri 1/p , ∀x ∈ Rn , for a constant • a homogeneous p-norm is defined as xΔ,p = i=1 |xi | p ≥ 1. For simplicity, we choose p = 2 and write xΔ for xΔ,2 . Lemma A.1 [22] Let f be a continuous vector field on Rn such that the origin is a locally asymptotical stable equilibrium point. Assume that f is homogeneous of degree k for some r ∈ (0, +∞)n . Then, for any p ∈ N ∗ and any m > p · max{ri }, there exists a strict Lyapunov function V for system (A1), which is homogeneous of degree m and of class C p . As a direct consequence, the time derivative V˙ is homogeneous of degree m + k. Lemma A.2 [25] For a continuous system x˙ = f (x), f (0) = 0, x ∈ Rn
(A2)
suppose that there exists a C 1 positive definite and proper function V : Rn → R and real numbers k > 0 and α ∈ (0, 1) such that V˙ + kV α is negative semi-definite. Then the origin is a finite-time stable equilibrium of (A2). Lemma A.3 [22] Suppose V : Rn → R is a homogeneous function of degree τ with respect to the dilation weight Δ. Then the following hold: (i) ∂V /∂xi is homogeneous of degree τ − ri with ri being the homogeneous weight of xi ; (ii) there is a constant c¯ such that V (x) ≤ c¯xτΔ . Moreover, if V (x) is positive definite, then there exists a constant c > 0, such that V (x) ≥ cxτΔ . Lemma A.4 Suppose c and d are two positive real numbers. Given any real-valued function γ(x, y) > 0, the following inequality holds |x|c |y|d ≤
c c d γ(x, y)|x|c+d + γ − d (x, y)|y|c+d . c+d c+d
(A3)
References [1] Mazenc F, Praly L, Dayawansa WP. Global stabilization by output feedback: examples and counterexamples. Systems and Control Letters 1994;23(2):119–125. [2] Qian CJ, Lin W. Output feedback control of a class of nonlinear systems: a nonseparation principle paradigm. IEEE Trans. Automat. Control 2002;47(10):1710–1715. [3] Qian CJ. A homogeneous domination approach for global output feedback stabilization of a class of nonlinear systems. In Proceedings of 2005 American Control Conference, Portland, OR, USA, pages 4708–4715, June 2005. [4] Andrieu V, Praly L, Astolfi A. Homogeneous approximation and recursive observer design and output feedback. SIAM J Control and Optimization 2008;47(4):1814–1850. 11
[5] Andrieu V, Praly L, Astolfi A. High gain observers with updated gain and homogeneous correction terms. Automatica 2009;45(2):422–428. [6] Zhai JY, Qian CJ. Global finite-time stabilization via output feedback for upper-triangular systems with unknown output gain. In Proceeding of 49th IEEE Conference on Decision and Control, Atlanta, USA, pages 4096–4101, December 2010. [7] Esfandiari F, Khalil H. Output feedback stabilization of fully linearizable systems. International Journal of Control 1992;56(5):1007–1037. [8] Isidori A. Nonlinear control systems (3rd edition). Springer-Verlag: Berlin;1995. [9] Khalil H, Esfandiari F. Semi-global stabilization of a class of nonlinear systems using output feedback. IEEE Trans. Automat. Control 1993;38(9):1412–1415. [10] Lin Z, Saberi A. Robust semi-global stabilization of minimum-phase input-output linearizable systems via partial state and output feedback. IEEE Trans. Automat. Control 1995;40(6):1029– 1041. [11] Teel A, Praly L. Global stabilization and observability imply semi-global stabilization by output feedback. Systems and Control Letters 1994;22(5):313–325. [12] Qian CJ. Semi-global stabilization of a class of uncertain nonlinear systems by linear output feedback. IEEE Trans. Circuits Syst II Express Briefs 2005;52(4):218–222. [13] Shim H, Son YI, Seo JH. Semi-global observer for multi-output nonlinear systems. Systems and Control Letters 2001;42(3):233–244. [14] Lin W, Qian CJ. Semi-global robust stabilization of MIMO nonlinear systems by partial state and dynamic output feedback. Automatica 2001;37(7):1093–1101. [15] Zhai JY, Qian CJ, Ye H. Semi-global output feedback stabilization of a generalized class of MIMO nonlinear systems. ASME Journal of Dynamic Systems, Measurement, and Control 2011;133(6):064501.1–064501.5. [16] Zhai JY, Du HB. Semi-global output feedback stabilization for a class of nonlinear systems using homogeneous domination approach. ISA Transactions 2013;52(2):231–241. [17] Yu Y, Zhong Y. Semi-global robust output tracking for non-linear uncertain systems in strictfeedback form. IET Control Theory and Applications 2012;6(5):751–759. [18] Jia RT, Qian CJ, Zhai JY. Semi-global stabilisation of uncertain non-linear systems by homogeneous output feedback controllers. IET Control Theory and Applications 2012;6(1):165–172. [19] Polendo J, Qian CJ. A generalized homogeneous domination approach for global stabilization of inherently nonlinear systems via output feedback. International Journal of Robust and Nonlinear Control 2007;17:605–629. [20] Zhai JY, Fei SM. Global practical tracking control for a class of uncertain nonlinear systems. IET Control Theory and Applications 2011;5(11):1343–1351. 12
[21] Kurzweil J. On the inversion of Lyapunov’s second theorem on the stability of motion. Ann. Math. Soc. Transl. Ser. 1956;24(2):19–77. [22] Bacciotti A, Rosier L. Liapunov functions and stability in control theory. Springer;2005. [23] Hermes H. Homogeneous coordinates and continuous asymptotically stabilizing feedback controls. Differential equations, Volume 127, Dekker: New York;1991. [24] Polendo J, Qian CJ. Decentralized output feedback control of interconnected systems with high-order nonlinearities. In Proceedings of 2007 American Control Conference, New York City, USA, pages 1479–1484, July 2007. [25] Bhat S, Bernstein D. Finite-time stability of continuous autonomous systems. SIAM J Control and Optimization 2000;38(3):751-766. [26] Bhat S, Bernstein D. Finite time stability of homogeneous systems. In Proceedings of 1997 American Control Conference, Albuquerque, New Mexico, USA, pages 2513–2514, June 1997.
13
Highlights
ŹA new design scheme is developed by coupling the finite-time stabilization theory with the homogeneous domination approach. ŹA controller is constructed for each nominal system by adding one power integrator technique. ŹThe stability and convergence analysis are carried out using the homogeneous systems theory. ŹThe efficiency of the decentralized output feedback stabilizers is demonstrated by a simulation example.